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Theorem rankxpsuc 9900

Description: The rank of a Cartesian product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 9897 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)

**Hypotheses**
Ref	Expression
rankxplim.1	⊢ 𝐴 ∈ V
rankxplim.2	⊢ 𝐵 ∈ V

**Assertion**
Ref	Expression
rankxpsuc	⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)))

Proof of Theorem rankxpsuc Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unixp 6281	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
2	1	fveq2d 6896	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
3		rankuni 9881	. . . . . . . 8 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
4		rankuni 9881	. . . . . . . . 9 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
5	4	unieqi 4916	. . . . . . . 8 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
6	3, 5	eqtri 2756	. . . . . . 7 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
7	2, 6	eqtr3di 2783	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
8		suc11reg 9637	. . . . . 6 ⊢ (suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
9	7, 8	sylibr 233	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)))
10	9	adantl 481	. . . 4 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)))
11		fvex 6905	. . . . . . . . . . . . . 14 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ V
12		eleq1 2817	. . . . . . . . . . . . . 14 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ((rank‘(𝐴 ∪ 𝐵)) ∈ V ↔ suc 𝐶 ∈ V))
13	11, 12	mpbii 232	. . . . . . . . . . . . 13 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → suc 𝐶 ∈ V)
14		sucexb 7802	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ V ↔ suc 𝐶 ∈ V)
15	13, 14	sylibr 233	. . . . . . . . . . . 12 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → 𝐶 ∈ V)
16		nlimsucg 7841	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ V → ¬ Lim suc 𝐶)
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim suc 𝐶)
18		limeq 6376	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → (Lim (rank‘(𝐴 ∪ 𝐵)) ↔ Lim suc 𝐶))
19	17, 18	mtbird 325	. . . . . . . . . 10 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 ∪ 𝐵)))
20		rankxplim.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
21		rankxplim.2	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
22	20, 21	rankxplim2 9898	. . . . . . . . . 10 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵)))
23	19, 22	nsyl 140	. . . . . . . . 9 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 × 𝐵)))
24	20, 21	xpex 7750	. . . . . . . . . . . . . 14 ⊢ (𝐴 × 𝐵) ∈ V
25	24	rankeq0 9879	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
26	25	necon3abii 2983	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
27		rankon 9813	. . . . . . . . . . . . . . . 16 ⊢ (rank‘(𝐴 × 𝐵)) ∈ On
28	27	onordi 6475	. . . . . . . . . . . . . . 15 ⊢ Ord (rank‘(𝐴 × 𝐵))
29		ordzsl 7844	. . . . . . . . . . . . . . 15 ⊢ (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
30	28, 29	mpbi 229	. . . . . . . . . . . . . 14 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
31		3orass 1088	. . . . . . . . . . . . . 14 ⊢ (((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))))
32	30, 31	mpbi 229	. . . . . . . . . . . . 13 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
33	32	ori 860	. . . . . . . . . . . 12 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
34	26, 33	sylbi 216	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
35	34	ord 863	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 × 𝐵))))
36	35	con1d 145	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
37	23, 36	syl5com 31	. . . . . . . 8 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ((𝐴 × 𝐵) ≠ ∅ → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
38		nlimsucg 7841	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → ¬ Lim suc 𝑥)
39	38	elv 3476	. . . . . . . . . . 11 ⊢ ¬ Lim suc 𝑥
40		limeq 6376	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim suc 𝑥))
41	39, 40	mtbiri 327	. . . . . . . . . 10 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
42	41	rexlimivw 3147	. . . . . . . . 9 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
43	20, 21	rankxplim3 9899	. . . . . . . . 9 ⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))
44	42, 43	sylnib 328	. . . . . . . 8 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim ∪ (rank‘(𝐴 × 𝐵)))
45	37, 44	syl6com 37	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim ∪ (rank‘(𝐴 × 𝐵))))
46		unixp0 6282	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)
47	24	uniex 7741	. . . . . . . . . . . . 13 ⊢ ∪ (𝐴 × 𝐵) ∈ V
48	47	rankeq0 9879	. . . . . . . . . . . 12 ⊢ (∪ (𝐴 × 𝐵) = ∅ ↔ (rank‘∪ (𝐴 × 𝐵)) = ∅)
49	4	eqeq1i 2733	. . . . . . . . . . . 12 ⊢ ((rank‘∪ (𝐴 × 𝐵)) = ∅ ↔ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
50	46, 48, 49	3bitri 297	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
51	50	necon3abii 2983	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
52		onuni 7786	. . . . . . . . . . . . . . 15 ⊢ ((rank‘(𝐴 × 𝐵)) ∈ On → ∪ (rank‘(𝐴 × 𝐵)) ∈ On)
53	27, 52	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∪ (rank‘(𝐴 × 𝐵)) ∈ On
54	53	onordi 6475	. . . . . . . . . . . . 13 ⊢ Ord ∪ (rank‘(𝐴 × 𝐵))
55		ordzsl 7844	. . . . . . . . . . . . 13 ⊢ (Ord ∪ (rank‘(𝐴 × 𝐵)) ↔ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
56	54, 55	mpbi 229	. . . . . . . . . . . 12 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵)))
57		3orass 1088	. . . . . . . . . . . 12 ⊢ ((∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))) ↔ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵)))))
58	56, 57	mpbi 229	. . . . . . . . . . 11 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
59	58	ori 860	. . . . . . . . . 10 ⊢ (¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
60	51, 59	sylbi 216	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
61	60	ord 863	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim ∪ (rank‘(𝐴 × 𝐵))))
62	61	con1d 145	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim ∪ (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥))
63	45, 62	syld 47	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥))
64	63	impcom 407	. . . . 5 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥)
65		onsucuni2 7832	. . . . . . 7 ⊢ ((∪ (rank‘(𝐴 × 𝐵)) ∈ On ∧ ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
66	53, 65	mpan 689	. . . . . 6 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
67	66	rexlimivw 3147	. . . . 5 ⊢ (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
68	64, 67	syl 17	. . . 4 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
69	10, 68	eqtrd 2768	. . 3 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 ∪ 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
70		suc11reg 9637	. . 3 ⊢ (suc suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴 ∪ 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
71	69, 70	sylibr 233	. 2 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ (rank‘(𝐴 × 𝐵)))
72	37	imp 406	. . 3 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
73		onsucuni2 7832	. . . . 5 ⊢ (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
74	27, 73	mpan 689	. . . 4 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
75	74	rexlimivw 3147	. . 3 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
76	72, 75	syl 17	. 2 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
77	71, 76	eqtr2d 2769	1 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 ∨ w3o 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∃wrex 3066 Vcvv 3470 ∪ cun 3943 ∅c0 4319 ∪ cuni 4904 × cxp 5671 Ord word 6363 Oncon0 6364 Lim wlim 6365 suc csuc 6366 ‘cfv 6543 rankcrnk 9781

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-reg 9610 ax-inf2 9659

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-r1 9782 df-rank 9783

This theorem is referenced by: (None)

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